Curve Space: Classifying Curves on Surfaces

We design signatures for curves defined on genus zero surfaces. The signature classifies curves according to the conformal geometry of the given curves and their embedded surface. Based on Teichmüller theory, our signature describes not only the curve shape but also the intrinsic relationship between the curve and its embedded surface. Furthermore, the signature metric is stable, it is close to identity between surfaces sharing similar Riemannian geometry metrics. Based on this, we propose a surface matching framework: first, with curve signatures, we match the partitioning of two surfaces defined by simple closed curves on them; second, the segmented subregions are pairwisely matched and then compared on canonical planar domains.